Portal Frame Example

To illustrate and verify the proposed method it is applied to the portal frame shown in Figure 8, for which experimental test results are available in the literature (Liew et al 1997). In the analysis, Young’s modulus E = 200 GPa and shear rigidity G = 77 GPa. Residual stress in the members for bending and axial behaviour is taken as ar = 0.3ay, while for shearing behaviour xr = 0.05xy, where ay and xy are respectively the normal and shearing yield stresses for steel. From the published test data (Liew et al 1997), the properties of the beam are: cross-section area A = 4740 mm2, moment of inertia I = 55 47×104 mm4, plastic modulus Z = 485×103 mm3, normal yield stress oy = 345 MPa, and shear yield stress xy = 199 MPa (based on von Mises criterion). The properties for both of the columns are: area A = 7600 mm2, moment of inertia I = 6103×104 mm4, plastic modulus Z = 654×103 mm3, and yield stresses ay = 336 MPa and xy = 194 MPa. Semirigid connections are modeled by the adopted four-parameter model (Richard et al 1975), for which the parameter values are obtained from the test results as described in the following. Upon applying a curve-fitting technique (Liu 2005) to the moment-rotation test results for the beam-to-column connection C1 (Liew et al 1997), the model parameters in Eq. (6) were determined to be M0 = 79 kN-m, Rce = 7202 kN-m/rad, Rcp = 144 kN-m/rad, and у = 0.57; similarly, for the column-to-base connection C2, the model parameters were determined to be M0 = 148 kN-m, Rce = 24721 kN-m/rad, Rcp = 151 kN-m/rad, and у = 0.78.

To match with the experimental test setup, the loads for the analysis procedure proposed by this study are monotonically increased up to the collapse load level by incrementally changing the magnitude of the load parameter H, while maintaining the fixed proportional coefficients on the horizontal and vertical loads shown in Figure 8. The beam is divided into three elements, while each column is taken as one element. The load-deflection behaviour of joint 6 was found by the analysis to be as given by Curve 1 in Figure 9(a). Also shown in Figure 9(a) are the test results (Liew et al 1997) and those found using a refined plastic hinge (PHINGE) analysis method (Chen et al 1996). It can be seen that at lower loading levels (H < 60 kN), the load-deflection results found by this study and PHINGE method are in good agreement with each other and those obtained in the test. At higher loading levels, the results of the current study are slightly less than those of the PHINGE method because the latter does not account for elastic shear deformation as herein. Note that the results obtained by both analysis methods are significantly less than the test results at higher load levels. The proposed method predicted structure collapse at load level Hc = 74 kN, which is close to the value of 77 kN predicted by the PHINGE method, but both of these values are considerably less than the 99 kN value found as the limit load by the experimental test.

5.56 H

H-

Figure 8. Tested semirigid portal frame

It likely that the above noted discrepancy between experimental and analytical results is a consequence of the analysis methods adopting connection behaviour data determined from separate pilot experiments (Liew et al 1997) that is different from that for the actual connections in the frame itself. In fact, a good prediction compared to the test results is found by adjusting the data for the beam-to-column connection C1 to be: M0 = 111 kN-m, Rce = 4629 kN-m/rad, Rcp = 1099 kN-m/rad, and у = 0.97. The proposed analysis procedure then finds the load-deflection behaviour of joint 6 to be as defined by Curve 2 in Figure 9(a), which is observed to be in very good agreement with the experimental test results. The plastic behaviour shown in Figure 9(b) corresponds to Curve 2 in Figure 9(a). It is seen that the development of plasticity at member ends is not very significant. This is because the connections C1 and C2 respectively have ultimate moment capacities Mu = 133 kN-m and Mu = 151 kN-m, which are not that much greater than the yield moment capacities My = 100 kN-m and My = 134 kN-m of the beam and columns, respectively. In essence, the behaviour of the portal frame is governed by semi-rigid connection behaviour rather than member plasticity.

Also shown in Figure 9(a) are two special cases where the portal frame was analyzed taking some or all of the connections to be rigid. For Case 1 when both the beam-to-column and beam-to-base connections were taken to be rigid, it can be observed from the corresponding load-deflection behaviour that the limit deflection is only about one-fifth of that found for the case of semi-rigid connections, at a limit load level Hf = 143.3 kN. For Case 2 when the beam-to-column connections were assumed rigid while the column-to-base connections were taken to be pinned, which is a conventional situation in design, the corresponding load-deflection behaviour is close to that when the connections are all semi-rigid, with a frame limit load capacity Hf = 82.4 kN. The post-elastic behaviour of the frame at the limit state for the two cases is shown in Figure 10. From Figure 10(a) for the case of all rigid connections, four plastic hinges (i. e., 100% plasticity) form in the beam and right column, while the left column base undergoes 52% plasticity under combined axial force and bending moment. The formation of the fourth plastic hinge at node 4 occurs when the horizontal load Hf =

143.3 kN. At the same time, the frame fails due to inelastic instability signaled by the horizontal displacement of node 6 becoming infinitely large (i. e., the corresponding stiffness coefficient tends to zero and causes the structure stiffness matrix to become singular). From Figure 10(b) for the case of beam-to-column rigid connections and column-to-base pinned connections, the beam experiences more serious plastic deformation than the columns. The formation of the plastic hinge at the right end
of the beam occurs when the horizontal load reaches H= 77.7 kN. At limit load level Hf = 82.4 kN, the frame fails due to inelastic instability signaled by the horizontal displacement of node 6 becoming infinitely large (i. e., the same failure mode as for the rigid frame). Table 1 indicates the different degrees of member-end stiffness degradation for the scenarios when all connections are semi-rigid and the two cases where all or some of the connections are rigid.

3 J(44M80

Table 1

. Stiffness degradation factors

Member

Th nrl

Semirigid

Rigid

End

Initial: rc0

rc

rP

r

Case 1: r

Case 2: r

C13

E1

0.671

0.051

0.966

0.051

0.482

C26

E2

0.671

0.049

0.943

0.049

0.000

B34

E3

0.115

0.051

1.000

0.051

1.000

0.572

B56

E6

0.115

0.043

0.829

0.043

0.000

0.000